In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and is used in logic diagrams to indicate a logic negation between the external logic state and the internal logic state (1 to 0 or vice versa).
A weather watch, on the other hand, typically refers to an alert issued to indicate that conditions are favorable for the development of dangerous weather patterns, although the dangerous weather conditions themselves are not currently present.
Watch: Conditions are favorable for the severe weather event in or near the watch area. When a watch is announced, stay alert and be ready to take action. Watches include include Tornado, Severe Thunderstorm, Winter Weather and Flash Flood. Warning: Severe weather event is imminent or occurring in the warned area.